SLOCC classification of pure states has been under intensive research [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}. For three qubits, two genuine entanglement states, namely the \(|GHZ\rangle \)  and \(|W\rangle \)  states, have been distinguished and characterized by the
vanishing or not of the three-tangle [2]}. For four or more qubits,
the number of SLOCC classes is infinite. It is highly desirable to divide
these infinite SLOCC classes into a finite number of families. Central to
the issue is the criteria to determine which family an arbitrary state
belongs to. Various methods have been undertaken to tackle the
classification of four-qubit states, including those based
on Lie group theory [3]}, on hyperdeterminant [1]}, on inductive approach [4]}, on string theory [5]},
and on polynomials (algebraic) invariants
[6]}, [7]}, [8]}, [9]}, [10]}, [11]}. Recently, the Majorana
representation has been used for SLOCC entanglement classification of \(n\) -qubit symmetric states [12]}. For \(n\)  qubits, it is known that the \(|l,n\rangle \)  states (symmetric Dicke states with \(l\)  excitations) are
inequivalent to the \(|GHZ\rangle \)  state or the \(|W\rangle \)  state under SLOCC
[13]}. Therefore it is necessary to develop schemes to find other
genuine entangled states which are inequivalent to the \(|GHZ\rangle \) , \(|W\rangle \) , or \(|l,n\rangle \)  states.
